Compared to more recent OCW calculus videos, I found this to be better in terms of respecting the learner's intellect, presenting the whole proof rigorously and teaching the student to think a certain way.
Calculus Revisited: Single Variable Calculus | MIT OpenCourseWare - https://ocw.mit.edu/resources/res-18-006-calculus-revisited-...
Complex Variables, Differential Equations, and Linear Algebra - https://ocw.mit.edu/resources/res-18-008-calculus-revisited-...
Calculus Revisited: Multivariable Calculus | MIT OpenCourseWare - https://ocw.mit.edu/resources/res-18-007-calculus-revisited-...
It was still entirely relevant to today even though it was a few decades old as the fundamentals of computer science are still fundamental.
Hearing that intro music still brings a smile to my face.
I just happen to be relearning math right now as I dive deeper into data science and this is perfect timing. Going to watch this series once I get through my math proofs book ("Book of Proof" by Richard Hammack which I recommend to people getting into math https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472...).
I skimmed through some of these video lectures, I wonder if I even should write books for this. Or simply href.
Thank you for the share. Will be of great help to me come summer.
Edit: add smiley face.
A formal system's territory is connected together by rigor; but whence comes the formal system itself? That has to be imagined.
Could you please post the links to this videos.
There is a much better PDF at Project Gutenberg .
The Gutenberg PDF is only 1.9 MB, compared to 12 MB for the scanned image PDF.
The Gutenberg page for this book  also has a link to the LaTeX source for the PDF.
As much as this probably makes me sound like an audiophile, I actually prefer the raw scans over what may essentially be a reprint. They show all the blemishes, unofficial additions, and other marks that make the book look more "real" and give it character.
In this instance, the raw scan has a picture of the cover, as well as an interesting note handwritten near the beginning: "Property of Edward M Sumner" with an address. IMHO these sorts of historical artifacts are worth preserving too. I've come across scans with random notes, bookmark fragments, and newspaper clippings included, and it's always fun to ponder how they got there. (Who is this person and how did he get the book? Is he the one who scanned it? Etc.)
The problem with used books tho is that sometimes I find hair. Ugh.
The PDF is 1892715 bytes = 1.892715 MB. I rounded to two significant digits, giving 1.9 MB.
When Gutenberg says 1.8 MB they probably actually mean 1.8 MiB. 1892715 bytes = 1.80503368377685546875 MiB, or 1.8 MiB to two significant digits.
(I included the link in case anyone was interested in the quality and to prove I didn't just make up the number!)
Weirdly, I can telnet to port 80 & GET / on the ipv6 address just fine.
LaTeX is not an aesthetic cure-all. The original, at least, has been subjected to the critical eye of a typesetter. There is no disputing taste, but I will anyway. If you're using an e-paper device, btw, you can vectorize the text in scanned pdfs like this by applying "Clearscan" in Acrobat, or the equivalent (after which, it becomes readable).
For quick reference, a.co and amzn.com are official shorteners and considered safe. amzn.to links however are third-party and often used to conceal affiliate links.
Just click the book, then under "Share" click the email button, and you'll see..........
and I'm not talking about brushing up on my linear algebra, that comes later, I'm talking high school level mathematics, stuff that I've largely forgotten or didn't "get" first time round.
I've seen these "machine learning for hackers!" articles who try to dish out a bit of maths saying that's all you need, but I don't think you can escape the fact that sometimes you just need to start from the beginning and work your way up
A great book for brushing up/re-learning pre-calc is "Precalculus Mathematics in a Nutshell" by George F. Simmons (you can find it used cheap). He really boils it down to the essentials. For example here's how he opens his chapter on Trig:
"Most trigonometry textbooks have been written by people who appear to believe that the importance of the subject lies in its applications to surveying and navigation. Even though very few people become surveyors or navigators, the students who study these books are expected to undertake many lengthy calculations about the heights of flagpoles, the widths of rivers and the positions of ships at sea. The truth is that the primary importance of trigonometry lies in a completely different direction - in the mathematical description of vibrations, rotations, and periodic phenomena of all kinds, including light, sound, alternating currents and the orbits of the planets around the sun. What matters most in the subject is not making computations about triangles, but grasping the trigonometric functions as indispensable tools in science, engineering and higher mathematics. These functions and their properties are the sole subject matter of this chapter."
The entire book has that vibe. It's wonderful.
I bit the bullet and focused my first few semesters almost entirely on remedial math education, trying to fill in that gap. Because I now had a strong motivation, a little age, and focus, I managed to become a very good math student and even received the occasional award during my undergrad math education.
I've lost much of it, but I now know I can learn it, and many of the topics that had intimidated me at first turned out to be not a big deal once I got it. One day as a hobby I plan on starting over from algebra and work up again for fun.
Also, even in pretty advanced CS (at least in my field, security and, particularly, cryptography), most of the nuts-and-bolts math is high school math.
I feel the same way though, I've been looking back through some basic algebra on Khan Academy too because I feel maths is very much my weak point.
Since the 2nd program I ever wrote was a script to do my math homework for me, I laughed at her.
Then I failed to pass the first year of CS bachelor. Twice.
I am very glad to see this most excellent book here, and will add it to my collection.
Good luck with your studies, and never feel embarrassed to better yourself!
If I may similarly influence anyone here, for themselves or someone they know, to read Calculus made easy to supplement their calculus coursework, I will be happy to have paid the favor forward in some small way.
By the way, Kalid Azad may be our modern day Silvanus P Thompson. And he has better tools, which he wields masterfully, than just pen and paper. Recommended too.
I vividly remember cramming for a test my freshman year of college, not having things click, and the final Aha! when a semester of pain disappeared with a few visualizations. The contrast between how most classes presented the material and what actually worked for me was jarring, and I had to share what helped. I hope other people share what works for them, in any format they can.
EDIT: I'm working through Khan Academy with one of my kids...would like to find other resources as well.
Look for opportunities to expand what they're doing in school. For example, after my kids had covered place values in school, I taught them binary. Their knowledge of base 10 is so much more robust when they learn base 2 as well.
Point is, I bet you already know plenty to be that ideal resource you seek for them. Good for you for wanting to start early.
I love this book. What's the best way to learn a mathematical field? To discover it yourself, piece by piece!
I wish this were a series.
Time spent learning Calculus is worthwhile; and if nothing else, understand the fundamental theorem. Overwhelmingly impressive.
We teach math backwards. We have a population that can barely do 4th grade arithmetic and it is socially okay.
Children and people believe that decimal points are accurate and that fractions are abstract when the exact opposite is true. 1/3 of a pizza is real and 0.333333 is a fake number.
We also teach movement and change as a word problem i.e. a train leaves Chicago at 25 MPH and another train leaves Flint, MI at 35 MPH when will the trains meet? That answer is an estimation of an unattainable constancy but people believe it is logical conclusion.
Pre-Calculus (needs to be repackaged with the idea of the one consistent in our world is change) should be taught before Algebra and Geometry. Make math into something where people can truly understand abstract and concrete. People actually think calculus is a hyper abstract algebra when in fact it is putting math into real world solutions.
The number one problem is calculus is 100% dependent on the teacher. A great teacher will make this work and a lower skilled teacher can absolutely kill almost all learning.
The main text was Stewart (decided at the department level), but I was teaching the Honors section which provided a good opportunity for me to ask for something extra. I had my students read this book alongside Stewart, and write weekly short essays comparing the two approaches. Many of the students turned in some quite good writing.
This is an outstanding book.
I don't know why exactly, it just sounds modern.
And anyway, I confess I don't know the industry at all, but I doubt anyone would talk about 1914 as representing 'modern publishing'.
Well. Now I have to read the whole thing I guess.
The book also does appear to go out of its way to keep language simple. For example:
We call the ratio dx/dy "the differential coefficient of y with respect to x." It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy. Instead of being frieghtened we will simply pronounce a brief curse on the stupidity of giving long crack-jaw names; and, having relieved our minds, will go on to the simple thing itself, namely the ration dx/dy.
What one fool can do, another can.
-Ancient Simian Proverb
Monkey see, monkey do?
Because to them it's obvious. Most maths teachers are so far ahead of the students they forgot they once were students themselves.
I've had 3 different ones in high school and the difference was incredible. All the way from 'only the best students learn anything' to 'everybody earns at least a passing grade'.
Maths and physics were the classes where the quality difference between the teachers stood out the most.
And to them it's wrong! Much is said in this book which is difficult (but not impossible) to rigourously justify. It took centuries for calculus to be placed on a rigourous mathematical foundation; this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. The presentation here (in particular the idea that dx² is negligible) can be made rigourous through "non-standard analysis", but this 20th century development is less-known (in fact, unknown to many mathematicians) and perhaps even more difficult than real analysis.
Rigour is not necessary to understanding, and can even be fatal to understanding, but it's how mathematicians work, and generally mathematics is taught by mathematicians.
Thank you for posting this! Such things always bothered me in high school, seemed like approximations that ought to bite you in the behind at least in some corner cases. Another example from TLA:
> dy = 2cos(θ + 1/2 dθ) · sin 1/2 dθ
> But if we regard dθ as indefinitely small, then in the limit we may
neglect 1/2 dθ by comparison with θ, and may also take sin 1/2 dθ as being the same as 1/2 dθ. The equation then becomes:
> dy = 2cosθ × 1/2 dθ
This again seems like very sloppy and careless kind of approximation that ought to bite you in the back - but knowing there are just (supposed-to-be) intuitive non-rigorous methods, and that these have actual rigorous backing, somehow soothes me.
You can get away with dy and dx if you just think "I'm not writing integral signs because they're annoying" and remember the rules of working with integrals. This is how stochastic differential equations work -- they're not even differential equations because Brownian motion is not differentiable, they're just notation.
0) Intermediate value theorem (continuous function with image between b and c must be such that there is x such that f(x)=d for d between b and c. This can be first proved where d=0 -- it's the method of bisection by nested intervals, really -- and then generalized)
0.5) The actual definition of derivatives known to everyone.
1) Rolle's theorem, i.e. the Intermediate Value Theorem on first derivatives (if something goes up and down then it must have a critical pont).
2) Mean Value Theorem (Rolle's theorem but rotated)
3) Taylor's theorem.
> and generally mathematics is taught by mathematicians
So that leaves the door wide-open to a whole army of mathematicians to who it is unknown that are teachers.
And those are the ones for who all of the above does not apply and who still treat their 'entry level problems' as obvious to all their students even when the evidence strongly suggests that to the students it is not at all obvious.
And I'm writing this as a high school kid who was pretty good in math but totally lost interest due to such teachers (that, and computers were far more interesting).
But you're right, why can't things be like this? Then I think, there is a certain art to having the ability to explain something so easily.. props to reddit's elif subreddit (https://www.reddit.com/r/explainlikeimfive/)
Seems like a good update to a classic, but there are some in the reviews complaining about Gardner https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/...
One problem that teaching calculus has is that it's very dependent on having a solid foundation in other mathematics such as advanced algebra, trig, etc. In today's school systems that encourage gaming the system as students and teaching to the test as educators it's rare for most students to actually understand or be competent with material they've allegedly studied. When they hit something that starts off where they left off and builds upwards, if they have any weaknesses in that foundation it will show immediately and slow them down immensely.
Add on to that all the other problems of typical calculus instruction such as a desire to make it hard as a matter of protecting calculus education as a status symbol for "smart" people, the ability to ratchet up the difficulty arbitrarily through requiring memorization of a potentially infinite set of "trivia" (every trig. identity, every method of differentiation/integration, and so on), while generally not concentrating on the abstract concepts or the fundamentals.
"... I had a calculus book once that said, 'What one fool
can do, another can'..."
I have this in Chapter 9: 'The Smartest Man in the World' in
Feynman's book 'The Pleasure of Finding Things Out'.
the original is French, the Author Gustave Bessiere was an engineer, mathematician and inventor:
I don't think they are (yet) copyright free, though.
Less pleased, though not at _all_ surprised, that the book addresses 'fifth form boys', with no mention of girls.
My problem with advanced math is not so much the understanding of principles but the application of these to solve new problems creatively.
I was able to master partial differential equations and pass exams but was never able to apply what was learned to solve new problems which I found very frustrating and was what ultimately led me to not pursue a career in the field.
Love it, thank you for sharing
> Mathematics is the art of giving the same name to different things
> The art of doing mathematics consists in finding that special case which contains all the germs of generality
> The vast majority of us imagine ourselves as like literature people or math people. But the truth is that the massive processor known as the human brain is neither a literature organ or a math organ. It is both and more.
> Sometimes I think that creativity is a matter of seeing, or stumbling over, unobvious similarities between things - like composing a fresh metaphor, but on a more complex scale.
There is an ounce of truth in what you say -- metaphors can be abused to draw false conclusions. This does not mean one should cower from using them.
From "Real Mathematical Analysis" 1st edition, p. 9:
Metaphor and Analogy
In high school English, you are taught that a metaphor is a figure of speech in which one idea or word is substituted for another to suggest a likeness or similarity. This can occur very simply as in "The ship plows the sea." Or it can be less direct, as in "his lawyers dropped the ball." What gives a metaphor its power and pleasure are the secondary suggestions of similarity. Not only did the lawyers make a mistake, but it was their own fault, and, like an athlete who has dropped a ball, they could not follow through with their next legal action. A secondary implication is that their enterprise was just a game.
Often a metaphor associates something abstract to something concrete, as "Life is a journey." The preservation of inference from the concrete to the abstract in this metaphor suggests that like a journey, life has a beginning and an end, it progresses in one direction, it may have stops and detours, ups and downs, etc. The beauty of a metaphor is that hidden in a simple sentence like "Life is a journey" lurk a great many parallels, waiting to be uncovered by the thoughtful mind.
Metaphorical thinking pervades mathematics to a remarkable degree. It is often reflected in the language mathematics choose to define new concepts. In his construction of the system of real numbers, Dedekind could have referred to A|B as a "type-two, order preserving equivalence class", or worse, whereas "cut" is the right metaphor. It corresponds closely to one's physical intuition about the real line. See Figure 3. In his book, Where Mathematics Comes From, George Lakoff gives a comprehensive view of metaphor in mathematics.
An analogy is a shallow form of metaphor. It just asserts that two things are similar. Although simple, analogies can be a great help in accepting abstract concepts. When you travel from home to school, at first you are closer to home, and then you are closer to school. Somewhere there is a halfway stage in your journey. You know this, long before you study mathematics. So when a curve connects two points in a metric space (Chapter 2), you should expect that as a point "travels along the curve," somewhere it will be equidistant between the curve's endpoints. Reasoning by analogy is also referred to as "intuitive reasoning."
Moral: Try to translate what you know of the real world to guess what is true in mathematics.
- Cat, Jordi. "On Understanding: Maxwell on the Methods of Illustration and Scientiﬁc Metaphor" Studies In History and Philosophy of Science Part B32, no. 3 (2001): 395-441
- Derman, Emanuel. Models of Behaving Badly. New York, NY: Free Press, 2011
- Foer, J. Moonwalking with Einstein. NY: Penguin, 2011
- Lutzen, Jesper. Mechanistic Images in Geometric Form. NY: Oxford University Press, 2005
- Maguire, E.A., D.G. Gadian, LS. Johnsrude, C.D. Good, J. Ashburner, R.SJ. Frackowiak, and CD. Frith. “Navigatioanelated Structural Change in the Hippocampi of Taxi Drivers.“ Proceedings of the National Academy of Sciences 97, no. 8 (2000): 4398-403
- Maguire, E.A., ER. Valentine, J.M. Wilding, and N. Kapur. “Routes to Remembering: The Brains Behind Superior Memory." Nature Neurosciencee, no. 1 (2003): 90-95
- Rocke, AJ. Image and Reality Chicago, IL: University of Chicago Press, 2010
- Solomon, Ines. “Analogical Transfer and 'Functional Fixedness' in the Science Classroom." Journal of Educational Researd 87, no. 6 (1994): 371-77
This is a very small fraction of what goes on -
1) Many many professors have not written a/the book
2) You're not going to sell 1000000's of books through the 1 class you teach every semester. A few, yes. But lots, no.
Oh, what's that... you can't just republish instantly because electronic computers haven't been invented yet? Well, just iterate and do things that don't scale :)
The word is still used by countries that use the 'long scale', which has the nice property that a billion = (1 million)^2 and a quadrillion = (1 million)^4 etc. If you can count in greek this also means that an n-illion times an m-illion equals an (n+m)-illion.
Any insight as to why things went 'the wrong way'?